scholarly journals Can each number be specified by a finite text?

2020 ◽  
Vol 3 (1) ◽  
pp. 8
Author(s):  
Boris Tsirelson
Keyword(s):  
Natural Number ◽  
Algebraic Number ◽  
Rational Number ◽  
Level Sets ◽  
Infinite Hierarchy

Contrary to popular misconception, the question in the title is far from simple. It involves sets of numbers on the first level, sets of sets of numbers on the second level, and so on, endlessly. The infinite hierarchy of the levels involved distinguishes the concept of "definable number" from such notions as "natural number", "rational number", "algebraic number", "computable number" etc.

scholarly journals Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

scholarly journals Automorphy factors for a Hilbert modular group

1988 ◽  
Vol 30 (2) ◽  
pp. 231-236
Author(s):  
Shigeaki Tsuyumine
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Modular Group ◽  
Algebraic Number Field ◽  
Maximal Order ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Totally Real

Let K be a totally real algebraic number field of degree n > 1, and let OK be the maximal order. We denote by гk, the Hilbert modular group SL2(OK) associated with K. On the extent of the weight of an automorphy factor for гK, some restrictions are imposed, not as in the elliptic modular case. Maass [5] showed that the weight is integral for K = ℚ(√5). It was shown by Christian [1] that for any Hilbert modular group it is a rational number with the bounded denominator depending on the group.

scholarly journals On a Certain Function Analogous to log|η(z)|

1970 ◽  
Vol 40 ◽  
pp. 193-211 ◽  
Author(s):  
Tetsuya Asai
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Modular Group ◽  
Classical Case ◽  
Algebraic Number Field ◽  
View Point ◽  
Limit Formula

The purpose of this paper is to give the limit formula of the Kronecker’s type for a non-holomorphic Eisenstein series with respect to a Hubert modular group in the case of an arbitrary algebraic number field. Actually, we shall generalize the following result which is well-known as the first Kronecker’s limit formula. From our view-point, this classical case is corresponding to the case of the rational number field Q.

Models for Rational Number Bases

1975 ◽  
Vol 68 (2) ◽  
pp. 113-123
Author(s):  
Jean J. Pedersen ◽  
Frank O. Armbruster
Keyword(s):  
Natural Number ◽  
Rational Number ◽  
Binary Representation ◽  
Base System ◽  
Negative Number

We think it’s important that you have some background on how this article came to be written. One of the authors sometimes plays a game he invented, called “hats,” in which he puts on a funny hat and converts base-ten numerals into another base representation. For example, he puts on a beanie and calls for anyone to say a favorite number. Someone says “six.” He goes to the chalkboard and writes “110,” The object of the game is for the students to figure out what he is doing, Someone who figures it out says, “Gotcha!” And then that person gets to wear the hat until someone else says, “Gotcha!” (In case you haven’t “got it” yet, the beanie hat converts numbers to the binary representation.) Several hats are used in the course of play, and each represents a different number base, but it’s always a natural number base greater than one. Recently we began to wonder if perhaps you can have a negative number base system in which the base is not an integer. And if you can, what will the numerals look like?

Is the Euclidean Algorithm Optimal Among its Peers?

2004 ◽  
Vol 10 (3) ◽  
pp. 390-418 ◽  
Author(s):  
Lou Van Den Dries ◽  
Yiannis N. Moschovakis
Keyword(s):  
Natural Number ◽  
Rational Number ◽  
Time Complexity ◽  
Euclidean Algorithm ◽  
Worst Case ◽  
On Demand ◽  
Recursive Program ◽  
Remainder Function ◽  
Proper Formulation ◽  
Image Position

The Euclidean algorithm on the natural numbers ℕ = {0,1,…} can be specified succinctly by the recursive programwhere rem(a, b) is the remainder in the division of a by b, the unique natural number r such that for some natural number q,It is an algorithm from (relative to) the remainder function rem, meaning that in computing its time complexity function cε (a, b), we assume that the values rem(x, y) are provided on demand by some “oracle” in one “time unit”. It is easy to prove thatMuch more is known about cε(a, b), but this simple-to-prove upper bound suggests the proper formulation of the Euclidean's (worst case) optimality among its peers—algorithms from rem:Conjecture. If an algorithm α computes gcd (x,y) from rem with time complexity cα (x,y), then there is a rational number r > 0 such that for infinitely many pairs a > b > 1, cα (a,b) > r log2a.

scholarly journals On the maximal abelian ℓ-extension of a finite algebraic number field with given ramification

1978 ◽  
Vol 70 ◽  
pp. 183-202 ◽  
Author(s):  
Hiroo Miki
Keyword(s):  
Natural Number ◽  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Class Number ◽  
Algebraic Number Field

Let k be a finite algebraic number field and let ℓ be a fixed odd prime number. In this paper, we shall prove the equivalence of certain rather strong conditions on the following four things (1) ~ (4), respectively : (1) the class number of the cyclotomic Zℓ-extension of k,(2) the Galois group of the maximal abelian ℓ-extension of k with given ramification,(3) the number of independent cyclic extensions of k of degree ℓ, which can be extended to finite cyclic extensions of k of any ℓ-power degree, and(4) a certain subgroup Bk(m, S) (cf. § 2) of k×/k×)ℓm for any natural number m (see the main theorem in §3).

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